ABSTRACT
Exercise 11: To think about: Can one prove a statement about a particular language using that same language? Richard’s paradox is discussed at length in [400, pp. 1686-1688]; I recommend the reader to have a look at that article, from the beginning, to become familiar with the kinds of questions that arise surrounding this paradox. Richard’s paradox is also treated in Kleene’s Introduction to Metamathematics [316, p. 38]. Also see [274] and [275], popular readings for “self reference” in logic and logic in general. You might be drawn into learning things like Go¨del’s incompleteness theorem, a result which says, roughly, if a language is rich enough so as to be able to decide the truth (in a finite mechanistic way) of any statement in the language, then contradictions arise; it also says that, essentially, any consistent system is not complete enough so as to be able to prove (from within the system) all the truths expressible in that system. For example, second order logic (where quantification is allowed over subsets, rather than just individual elements) is not complete, and so there are statements in mathematics whose truth is never decidable. See, for example, Section 6.12 for the discussion of such a truth.
